Ultraspherical polynomials

Gegenbauer polynomials

Orthogonal polynomials $ P _ {n} ( x, \lambda ) $ on the interval $ [ - 1 , 1 ] $ with the weight function $ h ( x) = ( 1 - x ^ {2} ) ^ {\lambda - 1 / 2 } $; a particular case of the Jacobi polynomials for $ \alpha = \beta = \lambda - 1 / 2 $( $ \lambda > - 1 / 2 $); the Legendre polynomials $ P _ {n} ( x) $ are a particular case of the ultraspherical polynomials: $ P _ {n} ( x) = P _ {n} ( x , 1 / 2 ) $.

For ultraspherical polynomials one has the standardization

$$ P _ {n} ( x , \lambda ) \equiv \ C _ {n} ^ {( \lambda ) } ( x) = $$

$$ = \ \frac{( - 2 ) ^ {n} }{n!} \frac{\Gamma ( n + \lambda ) \Gamma ( n + 2 \lambda ) }{\Gamma ( \lambda ) \Gamma ( 2 n + 2 \lambda ) } ( 1 - x ^ {2} ) ^ {- \lambda + 1 / 2 } \times $$

$$ \times \frac{d ^ {n} }{d x ^ {n} } [ ( 1 - x ^ {2} ) ^ {n + \lambda - 1 / 2 } ] $$

and the representation

$$ C _ {n} ^ {( \lambda ) } ( x) = \ \sum _ { k= 0} ^ { [ n / 2 ] } ( - 1 ) ^ {k} \frac{\Gamma ( n - k + \lambda ) }{\Gamma ( \lambda ) k ! ( n - 2 k ) ! } ( 2 x ) ^ {n-} 2k . $$

The ultraspherical polynomials are the coefficients of the power series expansion of the generating function

$$ \frac{1}{( 1 - 2 x w + w ^ {2} ) ^ \lambda } = \ \sum _ { n= 0} ^ \infty C _ {n} ^ {( \lambda ) } ( x) w ^ {n} . $$

The ultraspherical polynomial $ C _ {n} ^ {( \lambda ) } ( x) $ satisfies the differential equation

$$ ( 1 - x ^ {2} ) y ^ {\prime\prime} - ( 2 \lambda + 1 ) x y ^ \prime + n ( n + 2 \lambda ) y = 0 . $$

More commonly used are the formulas

$$ ( n + 1 ) C _ {n+1} ^ {( \lambda ) } ( x) = \ 2 ( n + \lambda ) x C _ {n} ^ {( \lambda ) } ( x) - ( n + 2 \lambda - 1 ) C _ {n-1} ^ {( \lambda ) } ( x) , $$

$$ C _ {n} ^ {( \lambda ) } ( - x ) = ( - 1 ) ^ {n} C _ {n} ^ {( \lambda ) } ( x) , $$

$$ \frac{d}{dx} [ C _ {n} ^ {( \lambda ) } ( x) ] = 2 \lambda C _ {n-1} ^ {( \lambda + 1) } ( x) , $$

$$ C _ {n} ^ {( \lambda ) } ( \pm 1 ) = ( \pm 1 ) ^ {n} \frac{2 \lambda ( 2 \lambda + 1 ) \dots ( 2 \lambda + n - 1 ) }{n!\ } = $$

$$ = \ ( \pm 1 ) ^ {n} \left ( \begin{array}{c} n + 2 \lambda - 1 \\ n \end{array} \right ) . $$

For references see Orthogonal polynomials.

Comments

See Spherical harmonics for a group-theoretic interpretation. Ultraspherical polynomials are also connected with Jacobi polynomials by the quadratic transformations

$$ C _ {2n} ^ {( \lambda ) } ( x) = \ \textrm{ const } P _ {n} ^ {( \lambda - 1/2 , - 1/2) } ( 2x ^ {2} - 1) , $$

$$ C _ {2n+ 1 } ^ {( \lambda ) } ( x) = \textrm{ const } x P _ {n} ^ {( \lambda - 1/2, 1/2) } ( 2x ^ {2} - 1) . $$

See [a1] for $ q $-ultraspherical polynomials.

References

[a1]	R.A. Askey, M.E.H. Ismail, "A generalization of ultraspherical polynomials" P. Erdös (ed.) , Studies in Pure Mathematics to the Memory of Paul Turán , Birkhäuser (1983) pp. 55–78